An airplane flies at an altitude of 5 miles toward a point directly over an observer. Consider θ and x as shown in the following figure. Write θ as a function of x. ​ ​

Find an equation of the tangent line to the graph of  at the point . ​

Find the slope of the graph of the function at the given value. ​ at ​

Use the alternative form of the derivative to find the derivative of the function at . ​

A spherical balloon is inflated with gas at the rate of 500  cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is 40  centimeters? ​

Find  if . ​

Find the derivative of the function. ​ ​

Use implicit differentiation to find an equation of the tangent line to the ellipse  at . ​

What is the name of this mineral?

Extra Credit: What is the more specific name for the luster of this mineral?